A342467 -id:A342467 - cp's OEIS frontend

A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 13, 14, 25, 37, 47, 58, 89, 98, 126, 159, 188, 219, 276, 303, 362, 423, 478, 536, 633, 688, 781, 881, 973, 1068, 1211, 1301, 1443, 1589, 1724, 1866, 2066, 2202, 2396, 2598, 2790, 2986, 3250, 3439, 3699, 3967, 4219, 4480, 4819, 5071

Offset: 1

Derek Holton and Alex Holton, Feb 06 2025

Solutions differing by only rotation or reflections are not counted separately.

If the numbers do not need to be distinct and rotations and reflections are counted separately we get A019298(n-2). If the numbers do not need to be distinct but rotations and reflections do not count separately we get A006918(n-2). If the six numbers must be distinct and reflections and rotations count separately we get 6*a(n). - R. J. Mathar, Feb 27 2025

			The a(9) = 1 solution is:
       1
     5   6
   3   4   2

R. J. Mathar, Illustrations - Examples (2025)
Index entries for linear recurrences with constant coefficients, signature (-1,0,2,3,2,0,-3,-4,-3,0,2,3,2,0,-1,-1).

Cf. A342467, A380105.

A380853 := proc(n)
    -3412+2353*n+30*n^3-480*n^2+(135*n-900)*(-1)^n ;
    %-144*(2*A010891(n)+A010891(n-1)+2*A010891(n-2)) ;
    %-160*(17*A049347(n)+8*A049347(n-1)) ;
    %-360*A057077(n) ;
    %+480*(-1)^n*(A099254(n)-A099254(n-1)) ;
    %/720 ;
end proc:
seq(A380853(n),n=1..40) ; # R. J. Mathar, Feb 27 2025

a(n) = my(c=0, t); for(x=3, n-5, t=n-x; for(y=2, min(x-1, t-1), for(z=1, y-1, if(#Set([x, y, z, t-y, t-z, n-y-z])==6, c++)))); c; \\ Jinyuan Wang, Feb 07 2025

G.f.: x^9*(1 + 4*x + 8*x^2 + 16*x^3 + 18*x^4 + 18*x^5 + 15*x^6 + 10*x^7)/((1 - x)^4*(1 + 2*x + 2*x^2 + x^3)^2*(1 + x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6)). - Stefano Spezia, Feb 08 2025

A380105(n) = a(n)-a(n-3). - R. J. Mathar, Mar 13 2025

A342384 Irregular triangle T read by rows: T(n, k) is the number of n-th order magic triangles with magic constant equal to A285009(n) + k, with 0 < k <= 3*n - 5.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 0, 4, 6, 4, 0, 2, 18, 38, 71, 108, 115, 115, 108, 71, 38, 18, 155, 351, 695, 1067, 1475, 1815, 2007, 1815, 1475, 1067, 695, 351, 155, 1891, 4768, 9872, 15370, 22527, 30096, 35731, 37957, 37957, 35731, 30096, 22527, 15370, 9872, 4768, 1891

Offset: 2

Stefano Spezia, Mar 10 2021

			The triangle begins:
    0;
    1,   1,   1,    1;
    2,   0,   4,    6,    4,    0,    2;
   18,  38,  71,  108,  115,  115,  108,   71,   38,   18;
  155, 351, 695, 1067, 1475, 1815, 2007, 1815, 1475, 1067, 695, 351, 155;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..118 (rows 2..10)
Terrel Trotter, Normal Magic Triangles of Order n, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32.
Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20.

Cf. A016777 (row length), A179805, A285009, A341740, A342467 (row sums).

\\ See A342467 for program code.
{ for(n=2, 6, print(A342384row(n))) } \\ Andrew Howroyd, Feb 05 2022

Terms a(14) and beyond from Andrew Howroyd, Feb 05 2022

A355119 a(n) is the number of order-n magic triangles composed of the numbers from 1 to n(n+1)/2 in which the sum of the k-th row and the (n-k+1)-st row is the same for all k and all three directions, counted up to rotations and reflections.

Original entry on oeis.org

1, 1, 0, 0, 7584, 5546793216

Offset: 1

Donghwi Park, Jun 19 2022

The magic sum is (n(n+1)/2 + 1)(n+1)/2.
For n >= 3, a(n) is a multiple of 6 because the rotation of only three corners does not affect the sum of the 1st row and n-th row.
This magic triangle is an analog of magic triangles from St. Olaf College, which are published in the Pi Mu Epsilon Journal (Fall 2021). Their magic triangles use square numbers of triangles.

			a(1) and a(2) are trivially 1.
a(3) is trivially 0 because the sum of 2nd row cannot be same for each direction.
a(4k) for positive integers k is trivially 0 because the magic sums are not integers in this cases.
An example of a solution at n=5:
         4
       7   9
     12  1  11
   14  2   3  13
  6  15  10  8  5
An example of a solution at n=6:
          9
        20 18
      21  8  13
    11   3  2  19
   10  6  4  7   12
 1  16  17 15  14  5

Gabriel Hale, Bjorn Vogen, and Matthew Wright, Magic Triangles, The Pi Mu Epsilon Journal (Fall 2021).
Donghwi Park, Source code for a(5)
Donghwi Park, Source code for a(6)

Cf. A000217, A006052, A008586, A342467, A356643, A356808.

a(n) = 0 if n is a multiple of 4. - Stefano Spezia, Jun 20 2022

a(6) from Donghwi Park, Dec 31 2023

A356643 a(n) is the number of order-n magic triangles composed of the numbers from 1 to n(n+1)/2 in which the sum of the k-th row and the (n-k)-th row is same for all k and all three directions, counted up to rotations and reflections.

Original entry on oeis.org

1, 0, 0, 0, 612, 22411, 0

Offset: 1

Donghwi Park, Aug 19 2022

The magic sum is n*(n*(n + 1)/2 + 1)/2.

			a(1) is trivially 1.
a(2) is trivially 0.
a(4k-1) for positive integers k is trivially 0 because the magic sums are not integers in these cases.
a(4) is 0.
An example of a solution at n=5:
.
                 1
             15     5
           9     4     7
       12     6     8    13
     3    11     2    10    14
.
An example of a solution at n=6:
.
                 5
             19    16
          12     1    20
        9     6    10     8
    18    11     7    21     2
  3    17    13     4    14    15
.

Donghwi Park, Source code for a(4)~a(6)

Cf. A000217 (triangular number), A006052 (magic square), A004767, A342467, A355119.

a(n) = 0 if n is a multiple of 4 minus 1.

A351223 a(n) is the number of triangular arrays containing the first 3*(n - 1) positive integers arranged with number n on each side and having different set of the sets of the side integers.

Original entry on oeis.org

1, 120, 7560, 369600, 15765750, 617512896, 22813670880, 807723671040, 27686621927250, 925166131890000, 30286238493551040, 974802747606105600, 30933063577681246800, 969808565876506272000, 30090926129273230320000, 925249170367839629537280, 28225069296255264089522250

Offset: 2

Stefano Spezia, Feb 05 2022

nonn

			a(2) = 1:
    1
   / \
  2 - 3
with the set of the sets of the side integers S = {{1, 2}, {1, 3}, {2, 3}}.

Cf. A000142, A000578, A342467.

Mathematica
```
Table[(3(n-1))!/(6((n-2)!)^3),{n,2,18}]
```

a(n) = (3*(n - 1))!/(6*((n - 2)!)^3).
With F the generalized hypergeometric function: (Start)
O.g.f.: x^2*F([4/3, 5/3, 2], [1, 1], 27*x).
E.g.f.: x^2*F([4/3, 5/3, 2], [1, 1, 3], 27*x)/2. (End)
a(n) ~ 3^(3*n-7/2)*n^2/(4*Pi). - Stefano Spezia, Dec 25 2024
D-finite with recurrence (n-2)^3*a(n) -3*(3*n-5)*(n-1)*(3*n-4)*a(n-1)=0. - R. J. Mathar, Feb 27 2025

A375416 Number of order n magic triangles composed of the numbers from 1 to n^2 in which the sum of each 2 X 2 subtriangle is the same, counted up to rotations and reflections.

Original entry on oeis.org

1, 4, 144, 38336, 539904, 87249024

Offset: 1

Donghwi Park, Aug 15 2024

An order n triangle contains binomial(n,2) upright 2 X 2 subtriangles and binomial(n-2,2) inverted 2 X 2 subtriangles. In total, there are n^2-3*n+3 subtriangles.
It seems that the sequence is likely finite. Considering each of the n^2! possibilities of arranging 1..n^2, for each of the (n^2-3n+3) subtriangles only one choice for the central value can give the magic sum. We should, therefore, divide (n^2)! by (n^2)^(n^2-3*n+3) to calculate an estimation of a(n). For n >= 16, (n^2)!/(n^2)^(n^2-3*n+3) < 1.
For n >= 3, a(n) is a multiple of 8, because swapping between a corner triangle and an edge-adjacent triangle generate different examples,
Disregarding corner swap, a(3) to a(6) would be "18, 4792, 67488, 10906128"

			a(1)=1 because there is only the trivial case without any subtriangle.
a(2)=4 because we can choose only the number in the central triangle.
a(3)=18, which is same for A342467(4)*8. Trotter's order 4 magic triangle can be transformed to this order-3 magic triangle disregarding corner swap.
For n = 3, numbers 1..9 are placed inside the triangles shown:
        o
       / \
      o-- o
     / \ / \
    o---o---o
   / \ / \ / \
  o---o---o---o
An example with magic sum=17:
        9
        5
      1   2
      6   4
   7    3    8
This corresponds to the magic perimeter triangle (A342467):
     1 9 5 2
      7   4
       6 8
        3

Donghwi Park, Source code for a(4).
Donghwi Park, Source code for a(5).

Cf. A006052, A342467, A355256.

cp's OEIS Frontend

A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.

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A342384 Irregular triangle T read by rows: T(n, k) is the number of n-th order magic triangles with magic constant equal to A285009(n) + k, with 0 < k <= 3*n - 5.

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A355119 a(n) is the number of order-n magic triangles composed of the numbers from 1 to n(n+1)/2 in which the sum of the k-th row and the (n-k+1)-st row is the same for all k and all three directions, counted up to rotations and reflections.

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A356643 a(n) is the number of order-n magic triangles composed of the numbers from 1 to n(n+1)/2 in which the sum of the k-th row and the (n-k)-th row is same for all k and all three directions, counted up to rotations and reflections.

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A351223 a(n) is the number of triangular arrays containing the first 3*(n - 1) positive integers arranged with number n on each side and having different set of the sets of the side integers.

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A375416 Number of order n magic triangles composed of the numbers from 1 to n^2 in which the sum of each 2 X 2 subtriangle is the same, counted up to rotations and reflections.

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